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Journal of Electrical Engineering 6 (2018) 193-205 doi:
10.17265/2328-2223/2018.04.001
Phase Modulation Characteristics of Spatial Light Modulator and
the System for Its Calibration
Zhang Gongjian1, Zhang Man1 and Zhao Yang2 1. Department of
Opto-Electronic Engineering, Chitose Institute of Science and
Technology, Chitose 066-8655, Japan
2. Xi’An Jiaotong University City College, No. 8715 Shangji
Road, Xi’an 710018, Shaanxi Province, China
Abstract: The phase modulation characteristics of liquid crystal
SLM (spatial light modulator) and the system for calibrating are
proposed. Michelson interferometer is employed for measuring the
modulation properties of device. A system interface for operating
SLM is designed with VC++ compiler. The initial distortion phase is
determined by measuring the reflective interference, and modulation
of device is compensated by using our system. By using the LUT
(lookup table) data provided by manufacture the wavelength disperse
calibration is also achieved successfully. Key words: liquid
crystal SLM, phase modulation, wavefront correction, adaptive
optics.
1. Introduction
The advent of SLMs (spatial light modulators) has contributed
enormously to various applications in photonics during the last
years. The device is regarded as a perfect wave-front controller
because of its advantages such as low-power consumption,
high-resolution, non-mechanical and programming control. In
general, SLM is applied to modulate amplitude, phase or
polarization of a light wave. However, reliable application of SLMs
as programmable diffractive optical elements requires a thorough
calibration, because the individual SLM from manufacture has its
inherent modulation property. Therefore, there are many problems
encountered in practical applications, such as the need for an
easy-to-use interface, an inexpensive and precise calibration and
measurement system, and a series of reliable and stable algorithms
for phase extraction. All of these are problems to be solved. In
view of these problems, this study attempts to explore and propose
a complete set of solutions to many problems encountered by SLM in
practical application from
Corresponding author: Gongjian Zhang, associate professor,
research fields: information photonics, electrical and optical
material science.
theoretical algorithm to experimental method. At first, the
phase modulation characteristics of liquid crystal SLM and the
system for calibrating are proposed. Michelson interferometer is
employed for measuring the modulation properties of device. A
system interface for operating SLM is designed with VC++ compiler.
Then, the initial distortion phase is determined by measuring the
reflective interference and modulation of device is compensated by
using our system. At the same time, by using the LUT data provided
by manufacture the wavelength disperse calibration is also achieved
successfully.
2. Mathematical Analysis and Modulation Manner
The experimental arrangement is shown in Fig. 1. It is based on
a Michelson interferometer adapted for polarization phase shifting.
With a He-Ne laser (λ = 632.8 nm) as the source, a spatially
filtered and expanded collimated beam is obtained in the usual
manner as shown in Fig. 1.
The polarizers P1, P2 and quarter wave plate are combined, so
that the reference beam (reflected from the mirror) and the sample
beam (reflected from the SLM) maintain their appropriate
polarization states on
D DAVID PUBLISHING
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194
Fig. 1 Optical test setup for analyzing SLM modulation
properties.
emerging from the cubic beam splitter. A polarizer P3 is
introduced to make the two beams interfere. CCD (Charge-coupled
device) is located in the back focal plane of L4. With this
arrangement the interferogram was recorded.
The test system is a reflective SLM that can be
equivalently unfolded in the form of a cascade of two identical
transmissive cells with mirror-symmetric structures. Using the
convenient Jones matrix formulation, the amplitude of the
propagating wave front reflected from the SLM after the final
polarizer can be expressed as Ref. [1].
exp 2 cos 2 sin 2 / 1 cos 2 /1 cos 2 / cos 2 sin 2 /
(1)
In this case, the complex amplitude reflectance and phase shift
of the entire devices are as followings:
cos 2 cos
1 cos 2 sin sin 2 cos (2)
2 tan1 cos 2 sin sin 2 cos
cos 2 cos (3)
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In general, the device is in both amplitude and phase modulated
of coupling manner. The phase only or most amplitude modulation can
be achieved by selecting an appropriate configuration. For example,
if ψ1 = ψ2 = 0, smaller β value range can be an amplitude
modulation, and larger β value will be in phase modulation.
In LCOS SLM, there are two modulation modes: amplitude and phase
modulation. In general, these two modes are in a coupled state.
However, if appropriate parameter conditions are selected, a simple
modulation mode with only amplitude modulation or phase modulation
can be obtained. Using Eqs. (2) and (3), as a function of parameter
β, we can obtain the transmittance T and the phase modulation
quantity δ. In Fig. 2, we show the dependence relation between the
transmittance T and the phase modulation measure δ. δ is relative
to the beta parameter (the polarization angle of incident light and
transmitted light is 0). It is important to note that the phase in
Eq. (3) is represented by the inverse trigonometric function, so
the obtained phase is often the wrapped phase. In order to obtain
the real phase, the wrapped phase should be unwrapped. As shown in
Fig. 2, in the case of smaller values of the beta (in the figure,
between 0 and square
root three PI/4), the phase change range is very small, the
transmission rate is basically changed linearly with the beta, and
so the amplitude modulation can be obtained in this range. On the
other hand, in the case of large beta (in the diagram, greater than
the square root of 3 PI/4), the change of the transmittance is
small, and the phase changes basically just follow the applied
voltage, so the modulation of this case will be the phase
modulation. If the parameters are set properly, the pure phase
modulation or pure amplitude modulation can be realized.
3. Experimental Principle
In this section, the algorithms for phase measuring, phase
extracting, and phase unwrapping will be summarized. All of these
functions will be integrated into this SLM operating
application.
3.1 Phase Recovery from Interferogram
(1) PSI (phase-shifting interferometry) method PSI method has
drawn much attention in many
areas of applications because of its high precision in optical
measurements. This method utilizes the temporal domain to collect a
series of interferograms by
Fig. 2 The relationship between phase modulation δ, transmission
T, and parameter β.
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adjusting the phase of the reference beam. To create multiple
interferograms, a method is needed to introduce the required phase
shift in the reference beam. The most commonly used method is a
movable mirror driven by a PZT (piezoelectric transducer) to induce
a path-length change in the reference beam. A similar effect is
achieved by translating a prism orthogonal to the beam. Other
methods which have also been used to introduce a phase shift
include translating diffraction grating perpendicularly to the
light beam and rotating polarization phase retarders. In all these
phase-shifting methods, the accuracy and stability of the motion of
the phase-shifter is critical to the accuracy of optical
measurements. Thus the accuracy and stability of the reference
phase-shifter poses one of the most serious limitations for
precision phase measurement in PSI systems.
(2) The Fourier transform method The optical flatness of
reflecting devices, such as
mirrors and LCOS-SLMs surface, can be easily measured by
interferometer. In optical measurement of the interferometer, the
interferogram fringe patterns are of the form I = a(x, y) + b(x,
y)cos(φr + φ0), here, φr and φ0 express the reference and initial
phase distributions respectively.
In order to get the quantitative evaluation of co-sinusoidal
fringe patterns obtained from an optical interferometer, the FFT
transformation method as the signal processing technique is
employed commonly.
Its advantage is that just single frame of interferogram is
needed for extracting the phase information.
In this case, usually a carrier was introduced into a normal
interference fringe pattern. It is similar to the carrier
modulation in the communication technology, here the examples as
usual are Moiré and Fringe Projection Techniques.
The interference patterns were then simulated from these modeled
phase values according to following expressions:
I x, y a , , cos , 2 (4)
where I(x, y) is the intensity distribution in the detection
plane (x, y), a(x, y) denotes the mean intensity of the
interference field, b(x, y) characterizes the modulation of the
detected interference signal, f = (fx, fy) is the spatial carrier
frequency, Nm and Na characterize multiplicative and additive noise
in the interference pattern. When the carrier frequency is large
enough, the interference fringes with opened loop are obtained. We
can use the Fourier transform to demodulate the interferogram and
extract the phase information.
Here, 1/2 , , , the symbol “*” means complex conjugate.
According to the basic properties of the linear and frequency shift
of the Fourier transform, you can get the spectrum as follows,
I u, v a , , , (5) where, a , is a narrow peak at the center of
the Fourier spectrum, representing the spectrum of the background
and the slowly varying amplitude. And
, and , are two spectra of the interferogram, which are complex
conjugates of each other, and they are symmetric relative to the
origin of frequency.
The phase extraction is to isolate the spectrum C or spectrum
C*, and to carry out the inverse Fourier transform, the package
phase of the interferogram can be obtained.
In summary, the Fourier transform method is based on the Fourier
transform of the distribution of the intensity of the interference
field. The two-dimensional Fourier transform of the interference
pattern is a Hermitian function and the amplitude spectrum thus
looks symmetric with respect to the dc-term. The spectral peak at
zero frequencies represents low frequency spectral component that
arises from the modulation of the background intensity of the
interferogram.
Two symmetric spectral side lobes carry the same
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information about the phase values φr. Using appropriate
adaptive band pass filters in the spatial frequency domain we can
extract one of side lobes. The wrapped phase values can be then
calculated by the inverse Fourier transform of the filtered
spectrum. However, there are still many difficult problems to
solve.
(i) Because being a wrapped phase, discontinuities of phase
values must be correctly unwrapped by appropriate mathematical
techniques.
(ii) In order to select appropriate spectrum and eliminate the
noise and background of interferogram, the filtering technique is
employed. However, the badly chosen filter may cause a coarse
disturbance of resulting data.
(iii) For the phase extraction from an interference pattern in
which carrier was introduced, it is necessary to eliminate the
phase tilt after phase recovery. This is because when the spectrum
is selected by filtering technology, even small errors in the
spectrum can cause relatively large phase tilt.
(3) RQPT (regularized quadrature and phase tracking) method.
As mentioned above, in the method of carrier interference fringe
measurement using FFT analysis, large errors may occur when the
fringe frequency is low. If the surface of the test is undulating,
it is not even possible to obtain the carrier interference
fringes.
In addition, in the fringe analysis method based on FFT, the
object of parsing is just the brightness of the interference fringe
information, in this way, because the target light phase and phase
conjugate and dc component exist at the same time, so still it
cannot get the required phase information directly.
It is necessary to separate the required information with the
help of filtering technology to extract the phase. Therefore, this
method is not guaranteed to be effective in most cases.
In addition, as the traditional optical interference measuring
method of phase shift method, it is changing the measured surface
and the relative
distance between the interferometer reference surface at the
same time, to capture multiple interference image, and to estimate
the surface shape from the information. However, it is difficult to
guarantee the stability of the instrument and reduce the cost of
the measuring instrument.
In this method, since it is necessary to capture a plurality of
images, there is a problem that accuracy deteriorates greatly in an
environment with disturbance such as vibration.
In view of the above, there is a growing need to measure the
phase accuracy from the one-shot interference pattern of the closed
fringe that is not restricted by the local frequency of the
fringes. In this section we discuss phase recovery from closed
fringe interference patterns.
In the past few years, lots of well-known techniques for phase
extraction from a single interferogram, which possess closed
fringe, have been developed [2-4]. One of the most successful
approaches is the RPT (regularized phase tracking) technique, which
is robust and powerful enough to demodulate many kinds of fringe
pattern, for example, both open and closed noisy fringe patterns.
In addition, it does not need further phase unwrapping processing.
The typical examples are as RPT,PIRPT,GRPT, RPT method, often fail
in the cases of complex interferograms and have relatively low
phase reconstruction accuracy. For the PIRPT method, it neglects
the fringe background and modulation term. For the method of
GRPT/iGRPT, because dependence of scanning path and fringe density
were improved, it was found that this method is efficient for
demodulating many patterns of fringe. The method in our research is
based on the RPT and GRPT/iGRPT. It was improved for unwrapping the
estimated phase. We take into account the following 3 factors:
(i) Initial iteration value setting, it can realize eliminating
ambiguity of the solution, and obtain a unwrapped phase;
(ii) Phase front approximation using Taylor
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Phase Modulation Characteristics of Spatial Light Modulator and
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198
polynomial up to third order differential, it may make the
iteration initial value setting more reasonable.
(iii) Optimal cost function consist of the terms of least square
optimization, regularization, and quadrature of fringe pattern,
making the cost function symmetry to improve the reliability so
that almost all fringe patterns including both open and closed
fringe pattern, could be demodulated and do not need any further
unwrapping processing.
3.2 Phase Unwrapping
In our operating system, it is possible to use a measured
distortion phase at a fixed wavelength combining with the LUT
provided by manufactory for the calibration of the wavelength
dispersion. However, the phase value obtained is usually wrapped
between –π and π. If one wants to use the LUT coefficient to
correct the wavelength disperse, the correct phase value is needed.
Next procedure is phase unwrapping processing [5-9]. The
relationship between wrapped phase ψ and unwrapped phase φ can be
expressed as ψ φ 2πk , in which k is integer. The relation can be
exchanged as following form.
, , , , 4 , , (6)
Here, the ρ , expresses the differential of the phase difference
at i, j position. Finally, a Poisson’s equation derived from this
differential scheme can be solved by a method of the fast discrete
cosine transform and an unwrapped profile of the phase distribution
can be obtained.
When the initial distortion phase of SLM is extracted based on
the Fourier spectrum analysis method of interference fringes, the
inclined plane is often introduced in the phase surface data. To
eliminate the tilt in this phase data, it is necessary to unwrap
the wrapped phase data.
In this case, the initiation process requires iteration.
However, the noise in the phase data can cause distortion in the
iteration calculation. This will greatly affect the unwrapping
precision. In this case, FFT or DCT (discrete cosine transform)
method has the function of noise filtering. When the iterative
calculation is not sufficient, the low-frequency components are
very distorted. When iterative computation is fully implemented
until it fully converges, the phase unwrapping tends to fail. In
order to achieve the goal of the successful solution, it is
necessary to eliminate the noise in the phase data and to
Fig. 3 The comparison of the unwrapping and rewrapping of
eight-helix surface.
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Phase Modulation Characteristics of Spatial Light Modulator and
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modify the surrounding area of the phase image. In other words,
it is necessary to remove the noise of large high frequency
components. In addition, as shown in the figure, if the pattern of
fringes is spirally distributed, the iterative method must be
applied to the above algorithm to get the correct results. It is
found that the discrete cosine transform based on least square
method is the most effective.
4. Experimental Results and Discussion
4.1 Application Interface of SLM Operation
A system interface for operating SLM is designed with VC++
compiler as shown in Fig. 4. All control mask images were BITMAP
format in the gray scale values as change in applied voltage. The
file menu includes the mask image open, save, recent used file
items. The initial setting function is in the setting items
including calibration background reset, and wavelength
compensation. First of all, mask images will be changed to in 8bit
gray scale. For the BITMAPFILEHEADER, only two items are needed to
revise:
bfOffBits=sizeof(BITMAPFILEHEADER)+sizeof(BITMAPFILEHEADER)+256*sizeof(RGBQUAD);
bfSize=bfOffBits+nImageSize; And for the BITMAPINFO, also there
are two
items needed to revise: bmiHeader.biBitCount=8;
miHeader.biSizeImage=nImageSize2; Other is Colormap, also needed to
rewrite. Advantages of SLM operation interface are as
following: can operate the SLM pixel by pixel from software
instead of hard ware operation; possible for calibrating the SLM
pixel by pixel.
4.2 SLM Initial Distortion Phase Measurement
The LCOS reflective SLM under analysis is manufactured by
Hamamatsu. The number of pixels is 1,280 × 1,024, and the total
size is 16 × 12.8 square millimeters. The size of each pixel is
12.5 × 12.5 square micrometers. The device allows 255 distinct
phase levels and a maximum phase variation of more than 2π can be
generated. So we measured the phase distortion
Fig. 4 The application interface of SLM.
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Phase Modulation Characteristics of Spatial Light Modulator and
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and combined the LUT data provided by manufacture for
calibrating the device. Fig. 1 shows the SLM test setup schema. The
incident light is aligned at an angle to the optic axis of the LC
modulator, which causes the SLM to vary the polarization angle of
the reflected light. With this configuration, polarization of the
incident light is polarized at a proper angle. One will observe the
reflected output light and the interferogram with a large contrast.
Adjusting the wave plate, the interference patterns were recorded
by CCD camera.
Fig. 5 shows the interference pattern recorded by the 4-step
phase-shifting method. The distortion phase can be obtained by
using phase equations of four interference patterns (3).
Fig. 6 shows the interference pattern obtained from
the carrier interference fringes and the initial distortion
phase obtained by the FFT analysis method. When correcting the
distortion phase, the photomap of the optical receiving part is
measured so as to accurately match the position in pixels (Fig.
7).
For the method of RQPT, we try to use a method, we call it
correlation-like analysis, dividing the whole area we want to
demodulate, into several sub-windows, performing the optimization
for every sub-window using above mentioned algorithm (see Section
3.1), the area in calculating is part overlapping with the
undetermined area, using correlation-like analysis, the data to
update in undetermined area could be a disambiguation solution. The
snapshot of the calculation process is shown in Fig. 8. The
experimental results show that the RQPT method has
Fig. 5 4-step phase shift interference fringes and wrapped phase
of SLM.
Fig. 6 Interference fringe pattern and phase extraction
result.
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Phase Modulation Characteristics of Spatial Light Modulator and
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Fig. 7 Photograph of SLM light receiving surface.
Fig. 8 QPT demodulate closed fringe pattern (sub-window overlap
correlation-like method).
the best effect. This can be verified by performance of the
correction of the SLM distortion phase, which will be discussed in
the next section.
4.3 SLM initial Distortion Phase Correction
When using an SLM as a diffraction optical element, because of
its initial phase distortion would be included in the CGH
(computer-generated holography) diffraction grating, if the initial
phase distortion is loaded into the SLM, interference fringe
distortion will be fixed. The calibration result of a light source
with a wavelength of 0.658 microns is shown in Fig. 9.
The first row is the interferogram with phase
distortion, and the second row is the interferogram after
distortion correction. For distorted interference fringes with
larger spacing (row 3), the calibration results are shown in Row
4.
4.4 Correction of SLM Wavelength Dispersion
In order to observe the change of the interference fringes with
wavelength, a phase mask with comparability of gray level is
adopted. This mask is shown in Fig. 10, with the phase of the left
half (or lower part) unchanged, and the phase of the right half (or
upper part) is gradual from 0 to 360 degrees.
For a configuration of which the phase jump line is
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Phase Modulation Characteristics of Spatial Light Modulator and
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(a) Interference fringes with smaller spacing, calibration
effect.
(b) Interference fringes of larger spacing, calibration
effect.
Fig. 9 Confirmation of calibration effect (wavelength 0.658
μm).
Fig. 10 Experimental result of SLM wavelength dispersion.
perpendicular to the interference fringes, if the maximum phase
modulation is 360 degrees, it will be
found that the numbers of the interference fringes where the
phase is gradual will be one more than one
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Phase Modulation Characteristics of Spatial Light Modulator and
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where the phase is constant, and the maximum dislocation of
interference fringes will locate in the center of the
interferogram.
For SLMs with a calibration wavelength of 633 nm, if the
wavelength used is greater than this reference wavelength, there
will be more than one interference fringe, and if the wavelength
used is less than this reference wavelength, there will be no more
than one increase in interference fringes in the phase-graded
region.
On the other hand, when the phase modulation maximum of the
spatial light modulator is PI, the maximum dislocation of
interference fringes will appear on the outer edge of the
interferogram.
The wavelength dispersion can be compensated well according to
the displacement of interference fringes caused by dispersion of
different wavelength.
In addition, the wavelength disperse compensation can also be
achieved by using the LUT data divided by manufacture. Fig. 11
shows the interfereogram before and after compensation. It is
obviously the initial wave front was compensated successfully.
4.5 Diffraction by Blazed Kinoform: The Generation of
Lageere-Gaussian Beam
Fig. 12 shows the distortion phase and two types of LG mask.
Fig. 13 shows the effect of correcting the
distortion phase when binary mask and phase modulation mask are
used for phase modulation SLM. Fig. 14 shows a photograph of the
intensity distribution of the vortex light wave generated by the LG
beam kinoform. From left, they are LG beam pattern without, and
with distortion initial phase calibration as well as the
interference between the order of +1 and -1. This is the
first-order diffracted light from the kinoform mask in the case of
mode index p and l both of 5, that is, mode LG .
In Fig. 15, it shows LG beam profile, comparison between
theoretical value and experimental value.
In the case where the parameter R shows the correlation between
the measured value and the theoretical value it is defined by the
sum of the square of the residual,
Q y y (7)and the sum of the squared values, ∑ .
R 1 Q/∑y (8)The parameter R representing the correlation,
gives
R = 0.6596. Here, y and y* represent measured values and
estimated values, respectively. The correlation coefficient between
the experimental value and the theoretical value was found to be
0.9417. As shown in Fig. 15, it can be seen that the experimental
results are in good agreement with the theoretical analysis.
Fig. 11 Phase compensation result.
Fig. 12 Distortion phase and LG mask.
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Phase Modulation Characteristics of Spatial Light Modulator and
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204
Fig. 13 Distortion corrected diffraction mask.
Fig. 14 LG beam pattern, comparison between theoretical value
and experimental value.
Fig. 15 LG beam profile, comparison between theoretical value
and experimental value.
5. Conclusion
The initial phase distortion caused by manufacture process and
silicon substrate surface error will largely affect the phase
modulation effect of device. We have measured this initial phase
distortion with reflective interferometer technique. For
characterizing properties of modulation and operating the device,
the system
interface for operating SLM is designed with VC++ compiler. With
this system, the initial distortion phase is determined by
measuring the reflective interference and modulation of device is
compensated by using our system.
For phase extraction from an interferogram, by using the method
of phase shifting or the method of Fourier transform, it was found
that the phase
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Phase Modulation Characteristics of Spatial Light Modulator and
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205
distortion although initial distortion and dispersion correction
of SLM are initially solved, there are some deviations. The reasons
can be as follows:
(1) For the four-step phase shift method, it is necessary to
record more than 4 interferograms, and the output power of the
laser light source will change during the measurement process,
resulting in a large measurement error. At the same time, the
mechanical vibration of the measurement system cannot be
ignored.
(2) Even in the case of carrier interference fringes, the
wavefront tilt of the reference interference light is usually
limited, and the bandwidth of the measured bands is limited to a
large extent. That is to say, because the fringe analysis method is
on the surface of the object under test, concave and convex change
frequency is lower than the frequency of the carrier fringes enough
premise established, so the high frequency component will become
less sensitive.
For this reason, an improved RQPT method was employed for
demodulating the fringes pattern. It was shown that the RQPT method
has the best effect. The advantages of SLM operation interface
provided in this study are as followings:
(1) It can operate the SLM pixel by pixel from software instead
of hardware operation.
(2) It is possible to calibrate the SLM pixel by pixel.
According to the displacement of interference fringes caused by
different wavelength dispersion, the wavelength dispersion can be
well compensated. Another way to calibrate wavelength dispersion is
to use the LUT data provided by the manufacturer to compensate for
the dispersion of the wavelength. In
practice, it is also found to be a very effective method. The
correction effect of the distortion phase is
verified by using the diffraction of computer-generated
holographic diffraction to produce LG beam. It can be seen that the
SLM diffraction after correction of phase distortion, the resulting
LG beam is in good agreement with the theoretical expectation.
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[3] Tian, C., Yang, Y., Liu, D., Luo, Y., and Zhuo, Y. 2010.
“Demodulation of a Single Complex Fringe Interferogram with a
Path-Independent Regularized Phase-Tracking Technique.” Appl Opt.
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[4] Kai, L., and Kemao, Q. 2012. “A Generalized Regularized
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[6] Shi, W., Zhu, Y., and Yao, Y. 2010. “Discussion about the
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